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Statistical and Topological Properties of Sliced Probability Divergences

arXiv 2003.05783 ↗First posted 12 Mar 2020stat.ML cs.LGcited by 83

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Abstract (arXiv)

The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between one-dimensional random projections of the two measures. However, the topological, statistical, and computational consequences of this technique have not yet been well-established. In this paper, we aim at bridging this gap and derive various theoretical properties of sliced probability divergences. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of a sliced divergence does not depend on the problem dimension. We finally apply our general results to several base divergences, and illustrate our theory on both synthetic and real data experiments.

Key facts. Cited by 83 papers in the map (data through 7 Oct 2026). First posted on arXiv 12 Mar 2020.

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Citing papers per year: 2021: 10; 2022: 9; 2023: 17; 2024: 16; 2025: 19; 2026: 12.

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  1. 2023Recent Advances in Optimal Transport for Machine Learningcited by 66
  2. 2024Learning Diffusion Priors from Observations by Expectation Maximizationcited by 60
  3. 2022Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distancescited by 41
  4. 2023Energy-Based Sliced Wasserstein Distancecited by 39
  5. 2021Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projectionscited by 33
  6. 2022Spherical Sliced-Wassersteincited by 31
  7. 2023Scalable Optimal Transport Methods in Machine Learning: A Contemporary Surveycited by 27
  8. 2023Quasi-Monte Carlo for 3D Sliced Wassersteincited by 22
  9. 2021Efficient Gradient Flows in Sliced-Wasserstein Spacecited by 21
  10. 2023Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signalscited by 19
  11. 2024Expected Sliced Transport Planscited by 19
  12. 2022Distribution Regression with Sliced Wasserstein Kernelscited by 18
38 more
  1. 2022Amortized Projection Optimization for Sliced Wasserstein Generative Modelscited by 17
  2. 2022Sliced Wasserstein Variational Inferencecited by 17
  3. 2022Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projectionscited by 16
  4. 2023Sliced Wasserstein Estimation with Control Variatescited by 16
  5. 2021Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applicationscited by 15
  6. 2024Sliced Wasserstein with Random-Path Projecting Directionscited by 15
  7. 2024A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Sciencecited by 14
  8. 2023Properties of Discrete Sliced Wasserstein Lossescited by 12
  9. 2023Markovian Sliced Wasserstein Distances: Beyond Independent Projectionscited by 11
  10. 2025Lightspeed Geometric Dataset Distance via Sliced Optimal Transportcited by 10
  11. 2021SLOSH: Set LOcality Sensitive Hashing via Sliced-Wasserstein Embeddingscited by 9
  12. 2024Sourcerer: Sample-based Maximum Entropy Source Distribution Estimationcited by 9
  13. 2023Robust Estimation under the Wasserstein Distancecited by 8
  14. 2023Scalable Infomin Learningcited by 8
  15. 2024Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variatescited by 8
  16. 2026Rectified LpJEPA: Joint-Embedding Predictive Architectures with Sparse and Maximum-Entropy Representationscited by 8
  17. 2021Statistical and Topological Properties of Gaussian Smoothed Sliced Probability Divergencescited by 7
  18. 2022Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distancescited by 7
  19. 2024Gaussian process regression with Sliced Wasserstein Weisfeiler-Lehman graph kernelscited by 7
  20. 2025Slicing the Gaussian Mixture Wasserstein Distancecited by 7
  21. 2025Differentiable Generalized Sliced Wasserstein Planscited by 7
  22. 2026MIND: Monge Inception Distance for Generative Models Evaluationcited by 7
  23. 2024Distributional Counterfactual Explanations With Optimal Transportcited by 6
  24. 2024Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributionscited by 6
  25. 2025KerJEPA: Kernel Discrepancies for Euclidean Self-Supervised Learningcited by 6
  26. 2021Sliced Multi-Marginal Optimal Transportcited by 5
  27. 2023Fast Optimal Transport through Sliced Wasserstein Generalized Geodesicscited by 5
  28. 2024Robust Distribution Learning with Local and Global Adversarial Corruptionscited by 5
  29. 2025Slicing Wasserstein Over Wasserstein Via Functional Optimal Transportcited by 5
  30. 2021Set Representation Learning with Generalized Sliced-Wasserstein Embeddingscited by 4
  31. 2023Convergence of SGD for Training Neural Networks with Sliced Wasserstein Lossescited by 4
  32. 2023Differentially Private Gradient Flow based on the Sliced Wasserstein Distancecited by 4
  33. 2024Summarizing Bayesian Nonparametric Mixture Posterior -- Sliced Optimal Transport Metrics for Gaussian Mixturescited by 4
  34. 2025Efficient Transferable Optimal Transport via Min-Sliced Transport Planscited by 4
  35. 2021Active Slices for Sliced Stein Discrepancycited by 3
  36. 2025Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysiscited by 3
  37. 2026Expanding SPHERE-JEPA: A Family of Statistical Regularizers for the Hyperspherecited by 3
  38. 2023PTLp: Partial Transport Lp Distancescited by 2

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